To see if I could do it, I tried to replace the CPU on my broken CPU board. At least now I know that this was a bit beyond my soldering skills. But anyway, since I had voided my warranty because I probed around inside my device, I thought I'd probe around a bit more. More specifically, I probed the SYSBOOT[15:0] pins when I had the CPU off the board. These determine the boot order amongst other things. Normally, the boot order is NAND, NANDI2C, MMC0 (which is the on board SD card), UART0 (which is the UART board on the short edge of the board. With jumper J16 installed, this changes into UART0, XIP (not present, so it is skipped), MMC0, NAND. So with J16 installed, you could probably boot one of the many available Linux images that run on the Sitara processor from the SD card...
Still trying to get a replacement CPU board though. It appears to be difficult.